Results of Tucker

Results of Cubatch are mainly graphical (to have main numerical results, see report in the menu file). Here is an overview of what is displayed after a Tucker model.

Mode i: Loadings of thecorresponding modes (1,2, or 3D plots)

weighted or scaled:

Let us suppose a 3-way array X of size [n,p,q].

Let us call A the 1st loading (matrix of size nxd) of a Tucker model and G the core (3-way array of size [d,e,f]):

• weighted: rows of loadings are projections of the corresponding cloud on an orthonoramal basis.Representation of the columns of svd(AGu,d).
• scaled: pure loading solution of Tucker (projection on non-orthonormal basis).Representation of the columns of A.

Tools:

• Process control (available for 2D plots)
  • Hotelling's T² ellipsoïd: confidence region.
  • Potential functions: trace an estimated density of the probability of the points of the plot.
• Quality index: represents only points which are sufficiently representative (i.e. good explained variation).

Format:

• Continuous: points are connected.
• Discrete: points are not connected.

• Integer.
• Label.
• Marker.

Preferences:

• Color
• Marker
• Line style
• Grid
• Visible: when ubchecked, the learning cloud becomes unvisible.
• options of tools:
  • Hotelling's T² ellipsoïd:
    • Confidence levels.
    • Size: number of dots for the ellipse.
  • Potential contour:
    • Confidence levels.
    • Size: number of dots for the ellipse.

Mixed modes

It is a representation of combinations of modalities (mode i x mode j): tools and preferences are similar to simple mode (see above).

Let us have an example:

Suppose Ai the loadings of a Tucker model of N-Way data of size R1x....xRn.

We are interested by the mixed mode 1x2, i.e. theR1xR2 modalities of the first modes.They can be seen as a cloud called C12 in a R3x...xRN dimensional space, but Tucker model finds a subspace of this one F12, which basis is orthonormal :coordinates of elements of this basis are the columns of:

(Kronecker product)

constrained or projection:

• projection: the projection of the elements of the cloud C12 in the subspace F12, which forms the cloud Cp12 (p for projection).
• constrained: Representation of the "constrained cloud" Cc12 on F12, it is the closest cloud of Cp12, but with some tensorial constraints:Coordinates of the points of the cloud are the rows of

Explained variance

Plot of the core and representation of the importance of n-uplets of tucker model.

Qualities of modalities

• Explained variance (angle)
• SSE (distance)
• T² Hotelling statistic (proximities of modalities comparate to the learning cloud).

Model

Representation of the Tucker model.

Residuals

Representation of the residuals of the Tucker model.